关于构造二次函数证明不等式,柯西不等式已知:ai>0 i=1,2,3...n.求证:(a1+a2+a3+...+an)(1/a1+1/a2+1/a3+...+1/an)≤n^2要有具体的,每一步的过程,谢,
来源:学生作业帮助网 编辑:作业帮 时间:2024/08/03 15:30:36
![关于构造二次函数证明不等式,柯西不等式已知:ai>0 i=1,2,3...n.求证:(a1+a2+a3+...+an)(1/a1+1/a2+1/a3+...+1/an)≤n^2要有具体的,每一步的过程,谢,](/uploads/image/z/3979175-23-5.jpg?t=%E5%85%B3%E4%BA%8E%E6%9E%84%E9%80%A0%E4%BA%8C%E6%AC%A1%E5%87%BD%E6%95%B0%E8%AF%81%E6%98%8E%E4%B8%8D%E7%AD%89%E5%BC%8F%2C%E6%9F%AF%E8%A5%BF%E4%B8%8D%E7%AD%89%E5%BC%8F%E5%B7%B2%E7%9F%A5%EF%BC%9Aai%3E0+i%3D1%2C2%2C3...n.%E6%B1%82%E8%AF%81%EF%BC%9A%28a1%2Ba2%2Ba3%2B...%2Ban%29%281%2Fa1%2B1%2Fa2%2B1%2Fa3%2B...%2B1%2Fan%29%E2%89%A4n%5E2%E8%A6%81%E6%9C%89%E5%85%B7%E4%BD%93%E7%9A%84%2C%E6%AF%8F%E4%B8%80%E6%AD%A5%E7%9A%84%E8%BF%87%E7%A8%8B%2C%E8%B0%A2%2C)
关于构造二次函数证明不等式,柯西不等式已知:ai>0 i=1,2,3...n.求证:(a1+a2+a3+...+an)(1/a1+1/a2+1/a3+...+1/an)≤n^2要有具体的,每一步的过程,谢,
关于构造二次函数证明不等式,柯西不等式
已知:ai>0 i=1,2,3...n.求证:(a1+a2+a3+...+an)(1/a1+1/a2+1/a3+...+1/an)≤n^2
要有具体的,每一步的过程,谢,
关于构造二次函数证明不等式,柯西不等式已知:ai>0 i=1,2,3...n.求证:(a1+a2+a3+...+an)(1/a1+1/a2+1/a3+...+1/an)≤n^2要有具体的,每一步的过程,谢,
应该是求证:[a(1)+a(2)+a(3)+······+a(n)][1/a(1)+1/a(2)+1/a(3)+······+1a(n)]≧n^2.
[证明]
构造二次函数:y=(√kx+1/√k)^2=kx^2+2x+1/k,其中k是正数.
显然有:kx^2+2x+1/k≧0.
依次令k=a(1)、a(2)、a(3)、a(4)、······、a(n),得:
a(1)x^2+2x+1/a(1)≧0,
a(2)x^2+2x+1/a(2)≧0,
a(3)x^2+2x+1/a(3)≧0,
······
a(n)x^2+2x+1/a(n)≧0.
将以上n个不等式左右分别相加,得:
[a(1)+a(2)+a(3)+······+a(n)]x^2+2nx+[1/a(1)+1/a(2)+1/a(3)+······+1a(n)]≧0.
令f(x)=[a(1)+a(2)+a(3)+······+a(n)]x^2+2nx+[1/a(1)+1/a(2)+1/a(3)+······+1a(n)]
∵k>0,∴a(1)+a(2)+a(3)+······+a(n)>0,
∴f(x)是一条开口向上的抛物线,
∴要满足f(x)≧0,就需要:
(2n)^2-4[a(1)+a(2)+a(3)+······+a(n)][1/a(1)+1/a(2)+1/a(3)+······+1a(n)]≦0,
∴[a(1)+a(2)+a(3)+······+a(n)][1/a(1)+1/a(2)+1/a(3)+······+1a(n)]≧n^2.
注:括号“( )”里的数字是下标.